Not a proof that aleph null and the order of the continuum are the same

Standard

Cantor’s diagonal arguement (Photo credit: Wikipedia)

One final point from Wheels, Life and Other Mathematical Amusements– this time a “non-proof”.

Some argue that the order of the counting numbers, $\aleph_0$ is the same as that of the continuum – in other words that there is no difference in the scale of these two infinities.

Here is an argument that is sometimes advanced in this way, I found it initially seductive, but the proof it is wrong is actually very simple.

We simple assign an integer to every member of the continuum by “reversing” its order so, for instance:

1        0.1
2        0.2
3        0.3
4        0.4
5        0.5
6        0.6
7        0.7
8        0.8
9        0.9
10      0.01
11       0.11

100    0.001
101     0.101

And so on…

So what is the proof that this isn’t a proof that destroys Georg Cantor‘s work and rocks modern mathematics to its foundations? Well, in essence it is a restatement, in a slightly different way, of our old friend the diagonalisation argument.

No matter how long this list goes on for no number on the left hand side will ever have $\aleph_0$ digits. Hence no number on the right will ever represent an irrational. Hence it is impossible to assign a counting number to all the members of the set of the continuum.

Think of $\frac{1}{3}$ – there could never be enough counting numbers to represent this as a decimal, as there will always be another 3 to add on at the end…

Don’t think about it too hard, though, as it bends the mind!